Integrable Deformations and Dynamical Properties of Systems with Constant Population

• Main Author: Cristian Lăzureanu
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Mathematics
• Subjects: Hamilton–Poisson systems; integrable deformations; Lotka–Volterra systems; Kolmogorov systems; stability; periodic orbits
• Online Access: https://www.mdpi.com/2227-7390/9/12/1378

In this paper we consider systems of three autonomous first-order differential equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi mathvariant="bold">x</mi><mo>˙</mo></mover><mo>=</mo><mi mathvariant="bold">f</mi><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">x</mi><mo>=</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mi mathvariant="bold">f</mi><mo>=</mo><mrow><mo>(</mo><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo><msub><mi>f</mi><mn>2</mn></msub><mo>,</mo><msub><mi>f</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>z</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is constant for all <i>t</i>. We present some Hamilton–Poisson formulations and integrable deformations. We also analyze the case of Kolmogorov systems. We study from some standard and nonstandard Poisson geometry points of view the three-dimensional Lotka–Volterra system with constant population.